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Theses and Dissertations

May 2016

Multi-Scale Modeling of Particle ReinforcedConcrete Through Finite Element AnalysisMir Zunaid ShamsUniversity of Wisconsin-Milwaukee

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Recommended CitationShams, Mir Zunaid, "Multi-Scale Modeling of Particle Reinforced Concrete Through Finite Element Analysis" (2016). Theses andDissertations. 1202.https://dc.uwm.edu/etd/1202

MULTI-SCALE MODELING OF PARTICLE REINFORCED CONCRETE

THROUGH FINITE ELEMENT ANALYSIS

by

Mir Zunaid Shams

A Thesis Submitted in

Partial Fulfillment of the

Requirements for the Degree of

Master of Science

in Engineering

at

The University of Wisconsin-Milwaukee

May 2016

ii

ABSTRACT

MULTI-SCALE MODELING OF PARTICLE REINFORCED CONCRETE THROUGH

FINITE ELEMENT ANALYSIS

by

Mir Zunaid Shams

The University of Wisconsin-Milwaukee, 2016

Under the Supervision of Professor Konstantin Sobolev

Concrete is the main constituent material in many structures. The behavior of concrete is

nonlinear and complex. Increasing use of computer based methods for designing and simulation

have also increased the urge for the exact solution of the problems. This leads to difficulties in

simulation and modeling of concrete structures. A good approach is to use the general purpose

finite element software, e.g ANSYS . Normal strength concrete is a composite material

represented by mechanically strong aggregates of various shapes and sizes incorporated into

weaker cementitious matrix. A number of simplified homogenized models have been reported in

the literature to represent the mechanical response of concrete. An accurate representation of the

spatial distribution of the aggregate particles is one of the most important aspects of real-scale

concrete modeling. A three-dimensional, numerical model, capable of predicting structural

reliability of concrete under various loading conditions has been developed. A micromechanical

heterogeneous model based on "real world" spatial distribution of aggregates was generated

using a packing algorithm. This model has been used to compute the stress-strain response of

iii

concrete by taking a representative cell homogenization approach. The results of numerical

analysis of this model were compared with existing models of particulate composite material.

The computational results demonstrate agreement within existing models and, therefore, can be

used for micromechanical modeling of composite material such as "real world" concrete

composites.

iv

TABLE OF CONTENTS

CHAPTER 2 16

2 Concrete Failures and Fracture 16

2.1 Multi-scale Modeling Basics 18

2.2 Multi-scale Finite Element Model (MsFEM) 20

2.2.1 Construction of Basis Functions 21

2.2.2 Global Formulation 22

CHAPTER 3 25

3 Computational Models 25

3.1 The Packing Algorithm 26

3.2 Finite Element Modeling 28

3.2.2 Lower Order Model with 3-D BEAM188 Elements 33

3.2.3 Structural Solid Shell Elements (SOLSH190) Model 35

3.2.4 Mixed Element Model for Model Order Reduction 36

CHAPTER 4 32

4 Results and Discussions 32

CHAPTER 5 39

5 Conclusions and Future Work 39

5.1 Summary of Results 39

5.2 Future Work 40

References 41

v

LIST OF FIGURES

Figure 1-1 Various simplified matrix/aggregate models for macro-mechanical

behavior

10

Figure 2-1 Typical Stress-strain curve for concrete and its constituents 17

Figure 2-2 Loading and unloading cycle of concrete 18

Figure 2-3 Schematic descriptions of coarse and fine grids 22

Figure 2-4 Basis functions example. Left: basis function with K being a

coarse element. Right: basis function with K being RVE

24

Figure 3-1 Engineering problem solving process 25

Figure 3-2 The simulation based on Sobolev packing algorithm: (a) two-

dimensional representation of the algorithm [19];the results of

two-dimensional packing with (b) K= -1 and (c) K= -3 [7]; (d)

particle size distribution for 3-dimensional packing of 300 and

10,000,000 aggregate particles

27

Figure 3-3 (a) Solid spheres generated in ANSYS from Sobolev packing

algorithm, (b) solid block representing the RVE size (c)

subtracting spheres outside the RVE (d) conformal finite element

mesh

31

Figure 3-4 Node numbering convention and coordinate system of SOLID65

element

32

Figure 4-1 1st principal stress contour plot of solid concrete block 33

Figure 4-2 2nd principal stress contour plot of solid concrete block 33

Figure 4-3 3rd principal stress contour plot of solid concrete block 34

Figure 4-4 Von Mises stress contour plot of solid concrete block 34

Figure 4-5 1st principal stress contour plot of aggregate packed concrete block 35

Figure 4-6 2nd principal stress contour plot of aggregate packed concrete

block

35

Figure 4-7 3rd principal stress contour plot of aggregate packed concrete block 36

Figure 4-8 Von Mises stress contour plot of aggregate packed concrete block 36

vi

LIST OF TABLES

Table 3-1 Material properties of cement mortar and aggregates 29

Table 1-2 Bending of stents inside arteries 9

Table 4-1 Stress result values for solid and aggregate filled concrete block 38

Table 2-1 Load vs elongation of the stent from experiment 17

vii

ACKNOWLEDGEMENTS

I would like to first and foremost thank my advisor, Dr. Konstantin Sobolev, for his

continuous support and guidance. He has been generously patient with me and my constant

questions.

I would like to express my gratitude to Dr. Ilya Avdeev for his valuable advices in several

stages in this research. It was a pleasure working with Reza Moini and Scott Muzenski. Their

contribution was vital for completing this thesis. I am grateful to Mehdi Gilaki who helped me in

Finite Element Modeling. It would be very difficult to complete this thesis on time without his

help.

I would like to appreciate all the help from Betty Warras at CEAS and Jenna Jazna at the

Graduate School.

Finally, I would like to thank my family for being so supportive and for being a ceaseless

inspiration to me.

1

CHAPTER 1

1 Introduction

The macro-mechanical material properties and behavior of portland cement and asphalt

concrete mixtures, such as mechanical strength, modulus of elasticity, creep parameters, and

shrinkage, greatly depend on the properties of their main constituent, the aggregates [1]. In addition

to material properties, the most important parameters affecting the macro-mechanical behavior of

particulate composite materials include packing density, compaction degree, particle size, and

spatial distribution of aggregates. It has been shown that aggregates packing methods and optimal

aggregate distribution can improve the mechanical response of concrete [1, 2].

Figure 1 depicts five classical models describing the micro-mechanical properties of

composites applicable to concrete. The closed form expressions that are derived from these models

for the homogenized modulus of elasticity (Ec) are summarized as follows [4, 6].

Figure 1-1: Various simplified matrix/aggregate models for macro-mechanical behavior [4, 6].

Parallel [4] mmppc VEVEE (1)

Series [4] m

m

p

p

c E

V

E

V

E

1

(2)

2

Hirsch [4]

m

m

p

p

mmppc E

V

E

VX

VEVEX

E1

11

(3)

Counto [4]

pm

p

p

m

p

c

EE

V

VE

V

E

2

1

2

1

2

1

1

111

(4)

Random (Maxwell Model) [6]

211

2121

m

p

m

p

p

m

p

m

p

p

mc

E

E

E

EV

E

E

E

EV

EE

(5)

In the equations (1) – (5), Vp is the volumetric proportion of particulate, Ep is the elastic

modulus of the particulate, Vm is the volumetric proportion of the matrix, and Em is the elastic

modulus of the matrix.

1.1 Motivation and Objectives

Engineered structures must be capable of performing their functions throughout a specified

lifetime while being exposed to a series of events that include loading, environment, and damage

threats. These events, either individually or in combination, can cause structural degradation,

which, in turn, can affect the ability of the structure to perform its function. The performance

degradation in structures made of composites is quite different when compared to metallic

components because the failure is not uniquely defined in composite materials. The size of the

material components is significantly smaller than the structural dimensions. The nonlinear

behavior of concrete can be attributed to the initiation, propagation, accumulation and coalescence

3

of micro cracks within the internal material structure. Thus, failure of concrete structures is a multi-

scale phenomena.

Since the beginning of the 1970s there has been a steadily increasing interest in developing

robust and reliable models that can simulate concrete cracking; much of the development is

certainly caused by the introduction of the finite element method and other numerical simulation

techniques in science and engineering.

Improvements in the computer models may eventually reduce the role of experiments.

Experiments are considered expensive, time consuming and cumbersome, only limited structural

conditions can be tested, some loading situations are so complex that testing is considered

impossible.

In this research, a Finite Element (FE) model was developed to analyze particle reinforced

concrete. A prior FE model of a solid non-reinforced concrete was used as a benchmark.

1.2 Literature Review

In a research article Nagai et. al. demonstrated a method of finite element mesh generation

in aggregate packed concrete. They pointed out that the difficulty in mesh generation comes from

the boundary of aggregate and mortar. They proposed a method where they used three-dimensional

digital image processing to generate mesh. They reported that they have developed the algorithm

of a new element for the aggregate-mortar interface zone [7].

A finite element simulation using ANSYS based on a test for compressive strength of

recycled coarse aggregate-filled concrete was presented by Jing et. al. in a research article

4

published in the Advanced Materials Research Journal. Opened crack and closed crack were

introduced through transfer coefficient in their finite element model [8]. However, the meshing

technique with the presence of aggregates in the specimen was not reported. Discussion about

handling the aggregate and mortar interface zone was also not available.

The method of generation of aggregates in a computational model is very important to

replicated the real world concrete. Leite et al. described a method of aggregate distribution with

the help of random number generation [9]. Their objective was to develop a mesocopic concrete

for simulating fracture processes.

He et. al. investigated the influence of aggregate packing on the elastic properties of

concrete [10]. They developed a numerical model where they packed triangular aggregates. In a

2D finite element model, each triangular aggregate particle had a corresponding circular aggregate.

Increase and decrease of Young’s Modulus and Poisson’s Ratio based on the change of area

fraction of aggregates were reported.

Use of lattice model for computational model of concrete is also found to be used by several

researchers [11]. Elias and Stang developed a 2D lattice model to demonstrate concrete aggregate

interlocking. The model is based on rigid cells interconnected by springs.

Gal and Kryvoruk demonstrated a two step homogenization method of fiber reinforced

concrete for multi-scale analysis [12]. Their homogenization approach includes first an analytical

method to homogenize the aggregates and their interfacial zones and then a numerical

homogenization algorithm is applied to homogenize the mortar, fibers and pre-homogenized

aggregates. In another research Wriggers and Moftah developed a finite element model of concrete

by incorporating aggregates, which were randomly generated using Monte Carlo's simulation

5

method [13]. Their meso-scale model was meshed using aligned approach for direct computation

of the concrete effective properties. They have reported a subsequent homogenization was required

during the numerical simulation.

Due to the highly non-homogenous nature of concrete it is necessary to analyze a concrete

model in multi-scale to fully understand the role of every component, such as, aggregates, voids,

reinforcing fibers etc. In his doctoral thesis, Kasai presented his work on multi-scale modeling of

concrete in a nano-scale [14]. The author found a difference of themechanical properties of solid

C-S-H between MD computation results and the nanoindentation-micromechanics combined

results. His explanation was that may be due tothe difference of the real structure of C-S-H and

the crystal structure of jennite andtobermorite . In another multi-scale approach, Choudhuri

utilized XEFG-method to analyze concrete material both on macroscopic and mesoscopic scale

[15]. Constituents of concrete, i.e. cement paste and aggregates, are modeled explicitly while

concrete is consideredas homogenized material on the macrscopic scale. Purpose of the research

was to represent concrete damage. non-linear isotropic damage model is used to describe the

tensile behavior of the cement paste. A cohesive crack model is introduced that can capture

complex mixed-mode fracture.

6

CHAPTER 2

2 Concrete Failures and Fracture

Concrete used in common engineering structures, basically is a composite material,

produced using cement, aggregate and water. Sometimes, as per need some chemicals and mineral

admixtures are also added. Experimental tests show that concrete behaves in a highly nonlinear

manner in uniaxial compression. Figure.1 shows a typical stress-strain relationship subjected to

uniaxial compression. This stress-strain curve is linearly elastic up to 30% of the maximum

compressive strength. Above this point tie curve increases gradually up to about 70-90% of the

maximum compressive strength. Eventually it reaches the pick value which is the maximum

compressive strength. Immediately after the pick value, this stress-strain curve descends. This part

of the curve is termed as softening. After the curve descends, crushing failure occurs at an ultimate

strain. A numerical expression has been developed by Hognestad [16], which treats the ascending

part as parabola and descending part as a straight line. This numerical expression is given as:

0 < 𝜀 < 𝜀0′ ,

𝜎

𝜎𝑐𝑢= 2

𝜀

𝜀0′ (1 −

𝜀

2𝜀0′ )

𝜀0′ < 𝜀 < 𝜀𝑐𝑢 ,

𝜎

𝜎𝑐𝑢= 1 − 0.15 (

𝜀 − 𝜀0′

𝜀𝑐𝑢 − 𝜀0′ )

7

Figure 2.1 Typical Stress-strain curve for concrete and its constituents [9]

The stress-strain curve for hardened cement paste is almost linear as shown in the figure. The

aggregate is more rigid than the cement paste and will therefore deform less (i.e. have a lower

strain) under the same applied stress. The stress strain curve of concrete lies between those of the

aggregate and the cement paste. However this relationship is non-linear over the most of the

range. The reason for this non-linear behavior is that micro-cracks are formed both at the

interface between aggregate particles and cement paste and within the cement paste itself.

Concrete taken through a cycle of loading and unloading will exhibit a stress-strain curve

as shown in the Figure 2.2. Concrete will not return to its original length when unloading mainly

due to creep and micro-crackling.

8

Figure 2.2 Loading and unloading cycle of concrete [9]

2.1 Multi-scale Modeling Basics

Many scientific and engineering problems involve multiple scales, particularly multiple

spatial or temporal scales or both. Spatial scale involves the physical size of the problem and

temporal scale involves the different constituents such as, composite materials, porous media etc.

Involvement either spatial or temporal scale increases the complexity of the problem by increasing

the size of computation. These types of problems can be efficiently handled by multi-scale

modeling. In most traditional multi-scale modeling the calculation of material properties or system

behavior on the macroscopic level is done using information or models from microscopic levels.

In other words it captures the small scale effect on the large scale, without resolving the small-

9

scale features. At present there is enormous interest in multi-scale approaches, where the behavior

of materials and structures is analyzed simultaneously at several different length-scales. The global

mechanical performance of materials is determined by the materials structures and the local

properties of constituents. The constituents also have their own material structures at a lower scale,

which determine their mechanical properties. Take cement-based materials as an example.

Concrete is a composite material consisting of coarse aggregates and mortar. The global

mechanical behavior of concrete is determined by the material structure of concrete and the

mechanical properties of coarse aggregates and mortar. At a lower scale mortar is made up with

sands in cement paste. The properties of mortar are related to the material structure of mortar, as

well as the local properties of sands and cement paste.

Homogenization method [17] and concurrent method [18] are usually employed to address

the failure modeling problem of heterogeneous materials. The homogenization method applies

when the scales can be ideally divided and separated, while the concurrent method is used when

the scales are somehow coupled. Generally speaking the homogenization method consumes less

computational resources, and the concurrent method provides more accurate simulation results.

The homogenization method requires that each block of heterogeneous material is taken

out and isolated to evaluate its mechanical properties, and then these properties are used as the

homogenized properties of the blocks and are put back to the network to simulate the global

performance of the original piece of material. The concurrent method demands that the connection

between neighboring blocks is preserved and the boundaries of blocks remain compatible with

each other during the simulation, the stress and strain also remain the same as if the domain was

not decomposed.

10

2.2 Multi-scale Finite Element Model (MsFEM)

Multi-scale Finite Element Model (MsFEM) typically works by capturing the multi-scale

structure of the solution via localized basis functions. Basis functions contain information about

the scales that are smaller than the local numerical. Basis functions are coupled through a global

formulation to provide a faithful approximation of the solution. MsFEM consists of two major

ingredients: multi-scale basis functions and global numerical formulation that couples these multi-

scale basis functions. Basis functions are formulated to incorporate localized multi-scale features

of the solution. A global formulation couples these basis functions to provide an accurate

approximation of the solution.

Mathematical formulation of MsFEM can be developed by considering a linear elliptical

equation as:

𝐿𝑢 = 𝑓 𝑖𝑛 Ω (2.1)

𝑢 = 0 𝑜𝑛 𝜕Ω

Where Ω is a domain in ℝ𝑑 (𝑑 = 2,3)

𝐿𝑢 = −𝑑𝑖𝑣(𝑘(𝑥)∇𝑢)

And 𝑘(𝑥) is a heterogeneous field varying over multiple scales.

It is additionally assumed that the tensor 𝑘(𝑥) = (𝑘𝑖𝑗(𝑥)) is symmetric and satisfies

𝛼|𝜉|2 ≤ 𝑘𝑖𝑗𝜉𝑖𝜉𝑗 ≤ 𝛽|𝜉|2, for all 𝜉 ∈ ℝ𝑑 and with 0<α<β

11

2.2.1 Construction of Basis Functions

The domain of interest is first decomposed into partitions called coarse grid and assume

that the coarse grid can be resolved through a finer resolution called fine grid. Let us consider that

Th be a usual coarse grid partition of Ω into finite elements (triangular, quadrilaterals, etc.).

Let xi be the interior nodes of Th and 𝜑𝑖0be the nodal basis of the standard finite element

space 𝑊ℎ = 𝑠𝑝𝑎𝑛𝜑𝑖0. For simplification, it is assumed that Wh consists of piecewise linear

functions if Th is a triangular partition. Definition of multi-scale basis function 𝜑𝑖is given by:

𝐿𝜑𝑖 = 0 in K, 𝜑𝑖 = 𝜑𝑖0 on ∂K, ∀𝐾 ∈ 𝑇ℎ, 𝐾 = 𝑆𝑖 (2.2)

Where 𝑆𝑖 = 𝑠𝑢𝑝𝑝𝜑𝑖0

These multi-scale basis functions coincide with standard finite element basis functions on the

boundaries of a coarse-grid block K, and are oscillatory in the interior of each coarse-grid block.

Computational domain smaller than coarse grid block K can be chosen if one can use smaller

regions (Kloc) to characterize the local heterogeneities within the coarse-grid block. Such regions

are called Representative Volume Elements (RVE). In this case the definition of multi-scale basis

function 𝜑𝑖 changes to:

𝐿𝜑𝑖 = 0 in Kloc, 𝜑𝑖 = 𝜑𝑖0 on ∂Kloc, ∀𝐾𝑙𝑜𝑐 ∈ 𝑇ℎ, 𝐾𝑙𝑜𝑐 = 𝑆𝑖 (2.3)

In general, the equation (2.2) is solved on the fine grid to compute basis functions. In Figure 2.3a

the basis function is consgtructed when K is a coarse partition element, and in Figure 2.3b the basis

function is constructed by taking K to be an element smaller than the coarse grid block size.

12

(a) (b)

Figure 2.3 Schematic descriptions of coarse and fine grids.

2.2.2 Global Formulation

The representation of the fine-scale solution through multi-scale basis functions allows

reducing the dimension of the computation. Then the approximation of the solution 𝑢ℎ = ∑ 𝑢𝑖𝜑𝑖𝑖

is substituted into the fine-scale equation. Here ui are the approximate values of the solution at

coarse grid nodal points. The resulting system is projected onto the coarse dimensional space to

find ui. This can be done by multiplying the resulting fine scale equation with coarse-scale test

functions.

Generalized MsFEM problem is then represented as:

Find uh = Vh such that

∑ ∫ 𝐾∇𝑢ℎ . ∇𝑣ℎ𝑑𝑥𝐾

= ∫ 𝑓𝑣ℎ𝑑𝑥Ω𝐾 ∀𝑣ℎ = 𝑉ℎ (2.3)

The above equation (2.3) is equivalent to Aunodal = b

Where A=(aij) with

13

𝑎𝑖𝑗 = ∑ ∫ 𝑘∇𝜑𝑖. ∇𝜑𝑗𝑑𝑥𝐾𝐾

Unodal = (u1, u2, . . . . , ui, . . . ) are the nodal values of the coarse scale solution and b = (bi) with

𝑏𝑖 = ∫ 𝑓𝜑𝑖Ω

Computation of aij requires the evaluation of the integrals using simple quadrature rule on the fine

grid.

MsFEM can be further generalized as:

Lu = f (2.4)

Where L: XY is an operator.

Multi-scale basis functions are replaced by multi-scale maps EMsFEM: WhVh

For each vhWh, vr,h = EMsFEMvh is defined as:

Lmapvr,h = 0 in K

Lmap captures the small scales.

Solution scheme of equation (2.4) can be:

Find ut,h Vh such that: ⟨𝐿𝑔𝑜𝑏𝑎𝑙𝑢𝑟,ℎ, 𝑣𝑟,ℎ⟩ = ⟨𝑓, 𝑣𝑟,ℎ⟩, ∀𝑣𝑟,ℎ ∈ 𝑉ℎ

Appropriate choises of Lmap and Lglobal are the essential part of MsFEM and guarantee the

convergence of the solution.

14

Figure 2.4 Basis functions example. Left: basis function with K being a coarse element. Right:

basis function with K being RVE. [12]

15

CHAPTER 3

3 Computational Models

Most engineering problems are solved using mathematical/numerical modeling.

Mathematical models are presented in terms of differential equations with a set of boundary and/or

initial conditions. Figure 3-1 shows a flow chart of engineering problem solving process.

Physical Problem Mathematical Model Numerical Model

- Differential equations

- Boundary conditions

- Initial conditions

- Finite Element Method

- Finite Difference Method

- Boundary Element Method

- Finite Volume Method

- Central Difference Method

- Meshless Method

Figure 3-1: Engineering problem solving process

Numerical models approximate exact solutions at discrete points of a given physical

system. FEA is a popular numerical technique to solve engineering problems because of its ability

to handle complex structures.

16

3.1. The Packing Algorithm

Given the relationship between input parameters and output material performance, it is of

great importance that the input parameters, including spatial distribution of aggregates, be highly

tunable. The Sobolev packing algorithm was used to create a parametric aggregate dispersion

for use in simulation of particulate composites (Figure 3.2a) [19-20].It was assumed that the

aggregates had spherical shape (diameters varying from Dmin to Dmax) and that there was no

contact between any two spheres within the predefined and controlled volume size. The packing

size distribution and volumetric ratio of aggregates to matrix are highly tunable and parametric-

based. This allows for future parameter coupling and output values for modeling real-scale

particulate composites (Figure 3a).The packing algorithm starts with randomly assigning the

location of the largest aggregate particle (Dmax). The packing continues by the Monte-Carlo

generation of the coordinates and subsequent placement of spheres with diameter di, wherein

Dmax≥ di ≥ Dmin. The center of the sphere must lie within the container and spheres must not

overlap. Any spheres that do not fit these criteria are discarded.

After achieving a certain number of packing trials, the minimum sphere diameter, Dmin, is

reduced as follows:

where Dmin is the minimum diameter, Dmax is the maximum diameter, K is the reduction

coefficient, and N is the number of trials.

17

a) b) c)

1

10

100

0.075 0.75 7.5

Sieve/Particle

Size, mm

Passing, %

10,000,000 aggregates

300 aggregates LimitsC33

0.55- power

d)

Figure 3.2 The simulation based on Sobolev packing algorithm: (a) two-dimensional

representation of the algorithm [19];the results of two-dimensional packing with (b) K= -1 and

(c) K= -3 [7]; (d) particle size distribution for 3-dimensional packing of 300 and 10,000,000

aggregate particles.

This method yields a simple pseudo-dynamic packing algorithm that produces randomly

packed set of aggregates in a given container [19-20].This method is further expanded to allow

18

coefficient-driven particle spacing, and, therefore, allowing a more realistic suspension model to

be generated:

Where r is the radius of a new sphere, d is a minimum distance to the surface of any

already packed set of spheres, kdel is an initial coefficient that provides the separation between

spheres, S is a coefficient to change the size of separation, and m is the number of steps taken to

change the size (for simplification, it is assumed that m = N).

This packing process yields a favorable result for the modeling of aggregate based

composites when the spacing algorithm is applied, as demonstrated by Figure 2 [19-20]. It

should be noted that the ultimate selection of the best aggregates. mix should be based on the

workability requirements, segregation potential, strength and stiffness equirements achieved at

the highest packing density possible [21]. Aggregate optimization can enhance the compressive

strength by identification of the best blend through multiple criteria [22].

3.2 Finite Element Modeling

A finite element model of the cube shaped matrix containing spherical aggregates was

created in ANSYS software. One set of generated 101 spheres were packed in an approximate

cube shaped control volume . The spheres were scaled to have a maximum diameter of 20 mm.

For the sake of simplicity, finer aggregates were not considered in the packing. A solid cube was

then created in such a way that it accommodates all the spheres and at the same time it replicates

a continuum of large scale pack . Outermost spheres were only partially contained inside the cube.

The portion of those outermost spheres was then subtracted by boolean operation to flush the cut

19

surface of spheres to the outer surface of the cube. Another Boolean operation called glue was

performed to tie the spheres to the cube so that they do not deform independently and freely under

pressure.

The volumes were then meshed with ANSYS Solid65 concrete elements. Uniaxial pressure

was applied to the block from the top surface while the bottom surface was constrained in all

degrees of freedom. No slip condition of the top and bottom surfaces was considered. Material

properties of the aggregate and mortar considered in this analysis are shown in Table 3.1

Table 3.1 Material properties of cement mortar and aggregates. [21]

Elastic Modulus Poisson’s Ratio

Mortar/matrix 25 GPa 0.29

Aggregates 75 GPa 0.29

20

(a) (b)

(c)

(d)

Figure 3.3: (a) Solid spheres generated in ANSYS from Sobolev packing algorithm, (b) solid

block representing the RVE size (c) subtracting spheres outside the RVE (d) conformal finite

element mesh.

3.2.1 SOLID65 3-D Solid Element

SOLID65 is used for the three-dimensional modeling of solids with or without reinforcing

bars. The solid is capable of cracking in tension and crushing in compression. In concrete

applications, for example, the solid capability of the element may be used to model the concrete

while the rebar capability is available for modeling reinforcement behavior. Other cases for which

the element is also applicable would be reinforced composites (such as fiberglass), and geological

21

materials (such as rock). The element is defined by eight nodes having three degrees of freedom

at each node: translations in the nodal x, y, and z directions. Up to three different rebar

specifications may be defined. The most important aspect of this element is the treatment of

nonlinear material properties. The concrete is capable of cracking (in three orthogonal directions),

crushing, plastic deformation, and creep.

.

Figure 3-4 Node numbering convention and coordinate system of SOLID65 element.

22

CHAPTER 4

4 Results and Discussions

Since concrete, by construction, is a brittle material that does not exhibit a defined yield

point. Moreover it has much lower strength is tension compared to the same in compression. So,

in order to analyze the stresses, the maximum principal stresses were of primary concern. For

comparison and boundary condition validation purposes the same boundary conditions were

applied to a solid concrete block of the same size. By looking at the resulting stress contour plots

it is found that the maximum stress zones resembles the actual laboratory test case. Figure 4.1,

4.2, 4.3 and 4.4 shows the 1st principal stress, 2nd principal stress, 3rd principal stress and von

Mises stress respectively for a solid concrete block.

In case of aggregate filled block the results of the stress contour are quite different from

the results of an assumed homogeneous block. In the aggregate filled block the higher stress

zones are located near and around the aggregates. Figure 4.5, 4.6, 4.7 and 4.8 shows the 1st

principal stress, 2nd principal stress, 3rd principal stress and von Mises stress respectively for

aggregate filled concrete block.

From the results of stresses presented in Table 4.1 it is found that the stresses increase

significantly with the inclusion of the aggregates.

23

Figure 4-1 1st principal stress contour plot of solid concrete block.

Figure 4.2 2nd principal stress contour plot of solid concrete block.

24

Figure 4.3 3rd principal stress contour plot of solid concrete block.

Figure 4.4 von Mises stress contour plot of solid concrete block.

25

Figure 4.5 1st principal stress contour plot of aggregate packed concrete block.

Figure 4.6 2nd principal stress contour plot of aggregate packed concrete block.

26

Figure 4.7 3rd principal stress contour plot of aggregate packed concrete block.

Figure 4.8 von Mises stress contour plot of aggregate packed concrete block.

27

Table 4-1 Stress result values for solid and aggregate filled concrete block.

1st Principal

Stress

2nd Principal

Stress

3rd Principal

Stress

Von Mises

Stress

Solid Concrete

Block 243 x 104 MPa 22 x 104 MPa 99 x 106 MPa 120 x 106 MPa

Aggregate

Filled

Concrete

Block

363 x 106 MPa 142 x 106 MPa 260 x 106 MPa 216 x 107 MPa

28

CHAPTER 5

5 Conclusions and Future Work

Finite element models of solid concrete block and concrete block with aggregate packing

were developed and analyzed for stresses. The constraints and loading conditions were validated

through the solution of solid concrete block by looking at its maximum stress zones. Same

boundary condition and loading were applied to the aggregate filled concrete block. As expected,

the location and distribution of the maximum stresses were different from the same in the solid

block. Results are summarized in the next sub-section..

5.1 Summary of Results

Results from the research are summarized below

Finite element stress results for the solid concrete block replicates the actual test results of

uniaxial compression on a concrete block. The higher stress zones are at the same location

where the failure occurs.

With inclusion of the aggregates the concrete block exhibits higher rigidity and

consequently it experiences higher stresses.

Higher stress occurs near the aggregates. It is because of the stress concentration at the

aggregate-mortar interface. It indicates that damage or fracture is likely to initiate at the

interface areas.

3-D solid-beam mixed element model shows lower stress than the solid only element model

under same amount of bending (Figure 4-11).

29

5.2 Future Work

Several major research directions can be recommended for a future development of the

finite element analysis of aggregate reinforced concrete:

1. Location of the crack initiation and nature of its propagation can be investigated through

cohesive zone modeling in FEA.

2. Air voids were neglected throughout this research, where air voids play a significant role in

concrete strength and fracture. Inclusion of air voids in the model will be a desired

improvement of this research.

3. Investigating different types of loading conditions may be another approach to simulate

different locations in a large structure.

30

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